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<TITLE>CSE 322 
Assignment 9 
Due Friday, March 1, 1996</TITLE>
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Assignment 9 
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<H1>CSE 322 
Assignment 9 
Due Friday, March 1, 1996</H1>
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<OL><LI>
Let <IMG  ALIGN=MIDDLE ALT="" SRC="img1.gif"> and
<IMG  ALIGN=MIDDLE ALT="" SRC="img2.gif"> be NFAs.
Construct a new NFA <b>M</b> that accepts the language
<P><IMG  ALIGN=BOTTOM ALT="" SRC="img3.gif"><P>
Think of <b>L</b> as the set of arbitrary shuffles of strings from <IMG  ALIGN=MIDDLE ALT="" SRC="img4.gif"> and
<IMG  ALIGN=MIDDLE ALT="" SRC="img5.gif">.  For example, if <b>aab</b> is in <IMG  ALIGN=MIDDLE ALT="" SRC="img6.gif"> and <b>bb</b> is in <IMG  ALIGN=MIDDLE ALT="" SRC="img7.gif"> then
the strings <b>aabbb, ababb, bbaab</b> and many other strings are in <b>L</b>.
Hint: The machine will have to run both <IMG  ALIGN=MIDDLE ALT="" SRC="img8.gif"> and <IMG  ALIGN=MIDDLE ALT="" SRC="img9.gif"> in parallel, but
only one at a time.  Thus, a cross product-like construction is in order.
<P>
You don't have to prove that your construction is correct, but you must
state a behavioral lemma which could be used in such a proof.
<P>
<LI>
Show that the set of all strings over the alphabet <IMG  ALIGN=MIDDLE ALT="" SRC="img10.gif"> with an
equal number of <b>a</b>'s as <b>b</b>'s is not a regular set.  You may use closure
properties of regular languages and the fact that the language
<IMG  ALIGN=MIDDLE ALT="" SRC="img11.gif"> is not regular.
<P>
<LI>
Show that the language <IMG  ALIGN=MIDDLE ALT="" SRC="img12.gif"> is not regular.  
You will have prove this by contradiction from first principles.
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<P><ADDRESS>
<I>James Fix <BR>
Tue Feb 27 08:16:35 PST 1996</I>
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